A Hybrid Controller for Obstacle Avoidancein an n-dimensional Euclidean Space

Soulaimane Berkane, Andrea Bisoffi, Dimos V. Dimarogonas

Abstract— For a vehicle moving in an n-dimensional Eu-clidean space, we present a construction of a hybrid feedbackthat guarantees both global asymptotic stabilization of a ref-erence position and avoidance of an obstacle correspondingto a bounded spherical region. The proposed hybrid controlalgorithm switches between two modes of operation: stabiliza-tion (motion-to-goal) and avoidance (boundary-following). Thegeometric construction of the flow and jump sets of the hybridcontroller, exploiting a hysteresis region, guarantees robustswitching (chattering-free) between stabilization and avoidance.Simulation results illustrate the performance of the proposedhybrid control approach for a 3-dimensional scenario.

I. INTRODUCTION

The obstacle avoidance problem is a long lasting problemthat has attracted the attention of the robotics and controlcommunities for decades. In a typical robot navigationscenario, the robot is required to reach a given destina-tion while avoiding to collide with obstacle regions in theworkspace. Since the pioneering work by Khatib [1] andthe seminal work by Koditscheck and Rimon [2], artificialpotential fields and navigation functions have been widelyused in the literature, see, e.g., [1]–[4], to deal with theobstacle avoidance problem. The idea is to generate anartificial potential field that renders the goal attractive andthe obstacles repulsive. Then, by considering trajectories thatnavigate along the negative gradient of the potential field, onecan ensure that the system will reach the desired target fromall initial conditions except from a set of measure zero. Thisis a well known topological obstruction to global asymptoticstabilization by continuous time-invariant feedback whenthe free state space is not diffeomorphic to a Euclideanspace, see, e.g., [5, Thm. 2.2]. This topological obstructionoccurs then also in the navigation transform [6] and (control)-barrier-function approaches [7]–[10].

To deal with such a limitation, the authors in [11] haveproposed a hybrid state feedback controller to achieve ro-bust global asymptotic regulation, in R2, to a target whileavoiding an obstacle. This approach has been exploited in[12] to steer a planar vehicle to the source of an unknownbut measurable signal while avoiding an obstacle. In [13], ahybrid control law has been proposed to globally asymptot-ically stabilize a class of linear systems while avoiding anunsafe single point in Rn.

This research was supported in part by the Swedish Research Council(VR), the European Research Council (ERC) through ERC StG BU-COPHSYS, the Swedish Foundation for Strategic Research (SSF), the EUH2020 Co4Robots project, and the Knut and Alice Wallenberg Foundation(KAW). The authors are with the Division of Decision and Control Sys-tems, KTH Royal Institute of Technology, SE-10044 Stockholm, Sweden.{berkane,bisoffi,dimos}@kth.se

In this work, we propose a hybrid control algorithm forglobal asymptotic stabilization of a single-integrator sys-tem that guarantees the avoidance of a non-point spher-ical obstacle. Our approach considers trajectories in ann−dimensional Euclidean space and we resort to tools fromhigher-dimensional geometry [14] to provide a constructionof the flow and jump sets where the different modes ofoperation of the hybrid controller are activated. Our proposedhybrid algorithm employs a hysteresis-based switching be-tween the avoidance controller and the stabilizing controllerin order to guarantee forward invariance of the obstacle-free region (related to safety) and global asymptotic stabilityof the reference position. The parameters of the hybridcontroller can be tuned so that the hybrid control lawmatches the stabilizing controller in arbitrarily large subsetsof the obstacle-free region. Preliminaries are in Section II,the problem is formulated in Section III, and our solutionis in Sections IV-V, with a numerical examplification inSection VI.

II. PRELIMINARIES

Throughout the paper, R denotes the set of real numbers,Rn is the n-dimensional Euclidean space and Sn is the n-dimensional unit sphere embedded in Rn+1. The Euclideannorm of x ∈ Rn is defined as ‖x‖ :=

√x>x and the geodesic

distance between two points x and y on the sphere Sn isdefined by dSn(x, y) := arccos(x>y) for all x, y ∈ Sn. Theclosure, interior and boundary of a set A ⊂ Rn are denotedby A,A◦ and ∂A, respectively. The relative complement ofa set B ⊂ Rn with respect to a set A is denoted by A\Band contains the elements of A which are not in B. Given anonzero vector z ∈ Rn\{0}, we define the maps:

π‖(z) := zz>

‖z‖2 , π⊥(z) :=In− zz>

‖z‖2 , ρ⊥(z) =In−2 zz

>

‖z‖2 (1)

where In is the n× n identity matrix. The map π‖(·) is theparallel projection map, π⊥(·) is the orthogonal projectionmap [14], and ρ⊥(·) is the reflector map (also called House-holder transformation). Consequently, for any x ∈ Rn, thevector π‖(z)x corresponds to the projection of x onto the linegenerated by z, π⊥(z)x corresponds to the projection of xonto the hyperplane orthogonal to z and ρ⊥(z)x correspondsto the reflection of x about the hyperplane orthogonal to z.For each z ∈ Rn\{0}, some useful properties of these mapsfollow:

π‖(z)z = z, π⊥(z)π⊥(z) = π⊥(z), (2)

π⊥(z)z = 0, π‖(z)π‖(z) = π‖(z), (3)

0

ε

ε′

c

B‖µc‖(µc)

µc

Fig. 1. The helmet region (dark grey) defined in (14).

ρ⊥(z)z = −z, ρ⊥(z)ρ⊥(z) = In, (4)

π⊥(z)π‖(z) = 0, π⊥(z) + π‖(z) = In, (5)

π‖(z)ρ⊥(z) = −π‖(z), 2π⊥(z)− ρ⊥(z) = In, (6)

π⊥(z)ρ⊥(z) = π⊥(z), 2π‖(z) + ρ⊥(z) = In. (7)

We define for z ∈ Rn\{0} and θ ∈ R the parametric map

πθ(z) := cos2(θ)π⊥(z)− sin2(θ)π‖(z). (8)

In (9)–(14), we define for v ∈ Rn\{0} some geometricsubsets of Rn, which are described below (14):

Bε(c) := {x ∈ Rn : ‖x− c‖ ≤ ε}, (9)L(c, v) := {x ∈ Rn : x = c+ λv, λ ∈ R}, (10)

P4(c, v) := {x ∈ Rn : v>(x− c)4 0}, (11)

C4(c, v, θ) := {x ∈ Rn : (x− c)>πθ(v)(x− c)40} (12)

= {x ∈ Rn : cos2(θ)‖v‖2‖x− c‖24(v>(x− c))2}C45(c, v, θ) := C4(c, v, θ) ∩ P5(c, v), (13)

H(c, ε, ε′, µ) := Bε′(c)\Bε(c)\B‖µc‖(µc), (14)

where the symbols 4 and 5 can be selected as 4 ∈ {=, <,>,≤,≥} and 5 ∈ {<,>,≤,≥}. The set Bε(c) in (9) is theball centered at c ∈ Rn with radius ε. The set L(c, v) in (10)is the 1−dimensional line passing by the point c ∈ Rn andwith direction parallel to v. The set P=(c, v) in (11) is the(n−1)−dimensional hyperplane that passes through a pointc ∈ Rn and has normal vector v. The hyperplane P=(c, v)divides the Euclidean space Rn into two closed sets P≥(c, v)and P≤(c, v). The set C=(c, v, θ) in (12) is the right circularcone with vertex at c ∈ Rn, axis parallel to v and aperture2θ. The set C4(c, v, θ) in (12) with ≤ as 4 (or ≥ as 4,respectively) is the region inside (or outside, respectively) thecone C=(c, v, θ). The plane P=(c, v) divides the conic regionC4(c, v, θ) into two regions C4≤ (c, v, θ) and C4≥ (c, v, θ)in (13). The set H(c, ε, ε′, µ) in (14) is called helmet andis obtained by removing from the spherical shell (annulus)Bε′(c)\Bε(c) the portion contained in the ball B‖µc‖(µc), seeFig. 1. The next geometric fact will be used.

Lemma 1 ([15]): Let c ∈ Rn and v1, v2 ∈ Sn−1 be somearbitrary unit vectors such that dSn−1(v1, v2) = θ for someθ ∈ (0, π]. Let ψ1, ψ2 ∈ [0, π] such that ψ1 + ψ2 < θ <π − (ψ1 + ψ2). Then

C≤(c, v1, ψ1) ∩ C≤(c, v2, ψ2) = {c}.Finally, we consider in this paper hybrid dynamical systems[16], described through constrained differential and differ-

ence inclusions for state X ∈ Rn:{X ∈ F(X), X ∈ F ,X+ ∈ J(X), X ∈ J .

(15)

The data of the hybrid system (15) (i.e., the flow set F ⊂ Rn,the flow map F : Rn ⇒ Rn, the jump set J ⊂ Rn, the jumpmap J : Rn ⇒ Rn) is denoted by H = (F ,F,J ,J).

III. PROBLEM FORMULATION

We consider a vehicle moving in the n-dimensional Eu-clidean space according to the single integrator dynamics:

x = u (16)

where x ∈ Rn is the state of the vehicle and u ∈ Rn is thecontrol input. We assume that in the workspace there existsan obstacle considered as a spherical region Bε(c) centeredat c ∈ Rn and with radius ε > 0. The vehicle needs toavoid the obstacle while stabilizing its position to a givenreference. Without loss of generality, we consider n ≥ 2 1

and take our reference position at x = 0 (the origin).Assumption 1: ‖c‖ > ε > 0.

Assumption 1 requires that the reference position x = 0is not inside the obstacle region, otherwise the followingcontrol objective would not be feasible. Our objective isindeed to design a control strategy for the input u such that:

i) the obstacle-free region Rn\Bε(c) is forward invariant;ii) the origin x = 0 is globally asymptotically stable;

iii) for each ε′ > ε, there exist controller parameters suchthat the control law matches, in Rn\Bε′(c), the law u =−k0x (k0 > 0) used in the absence of the obstacle.

Objective i) guarantees that all solutions of the closed-loopsystem are safely avoiding the obstacle by remaining outsidethe obstacle region. Objectives i) and ii), together, can not beachieved using a continuous feedback due to the topologicalobstruction discussed in the introduction. Objective iii) isthe so-called semiglobal preservation property [13]. Thisproperty is desirable when the original controller param-eters are optimally tuned and the controller modificationsimposed by the presence of the obstacle should be asminimal as possible. Such a property is also accounted for inthe quadratic programming formulation of [17, III.A.]. Theobstacle avoidance problem described above is solved via ahybrid feedback strategy in Sections IV-V.

IV. PROPOSED HYBRID CONTROL ALGORITHM FOROBSTACLE AVOIDANCE

In this section, we propose a hybrid controller thatswitches suitably between an avoidance controller and astabilizing controller. Let m ∈ {−1, 0, 1} be a discretevariable dictating the control mode where m = 0 correspondsto the activation of the stabilizing controller and |m| = 1corresponds to the activation of the avoidance controller,which has two configurations m ∈ {−1, 1}. The proposed

1 For n = 1 (i.e., the state space is a line), global asymptotic stabilizationwith obstacle avoidance is infeasible.

control input, depending on both the state x ∈ Rn and thecontrol mode m ∈ {−1, 0, 1}, is given by the feedback law

u=κ(x,m):=

{−k0x, m = 0

−kmπ⊥(x− c)(x− pm), m ∈ {−1, 1}(17)

where km > 0 (with m ∈ {−1, 0, 1}) and pm ∈ Rn(with m ∈ {−1, 1}, see (18) below) are design parameters.During the stabilization mode (m = 0), the control inputin (17) steers x towards x = 0. During the avoidance mode(|m| = 1), the control input in (17) minimizes the distance tothe auxiliary attractive point pm while maintaining a constantdistance to the center of the ball Bε(c), thereby avoiding tohit the obstacle. This is done by projecting the feedback−km(x − pm) on the hyperplane orthogonal to (x − c).This control strategy resembles the well-known path planningBug algorithms (see, e.g., [18]) where the motion plannerswitches between motion-to-goal and boundary-followingobjectives.

For the sets we now introduce, the reader is referred toFig. 2 for the rest of the section. For θ > 0 (further boundedin (22)), the points p1, p−1 are selected to lie on the cone2

C=≤(c, c, θ)\{c}:

p1 ∈ C=≤(c, c, θ)\{c} and p−1 := −ρ⊥(c)p1. (18)

Note that, by (18), p−1 opposes p1 diametrically with respectto the axis of the cone C=≤(c, c, θ) and also belongs toC=≤(c, c, θ)\{c} as per the next lemma.

Lemma 2 ([15]): p−1 ∈ C=≤(c, c, θ)\{c}.The logic variable m is selected according to a hybridmechanism that exploits a suitable construction of the flowand jump sets. This hybrid selection is obtained through thehybrid dynamical system{

x = κ(x,m)

m = 0(x,m) ∈

⋃m∈{−1,0,1}

Fm × {m} (19a){x+ = x

m+ ∈M(x,m)(x,m) ∈

⋃m∈{−1,0,1}

Jm × {m}. (19b)

The flow and jump sets for each mode m ∈ {−1, 0, 1} aredefined as (see (14) for the definition of the helmet H):

J0 := H(c, ε, εs, 1/2), (19c)

F0 := Rn\(J0 ∪ Bε(c)), (19d)

Fm := H(c, ε, εh, µ) ∩ C≥≤(c, pm − c, ψ), |m| = 1, (19e)

Jm := Rn\(Fm ∪ Bε(c)), |m| = 1, (19f)

see Fig. 2. The (set-valued) jump map is defined as

M(x, 0) :={m′∈{−1, 1} : x ∈ C≥(c, pm′− c, ψ)

}(19g)

M(x,m) := {0}, for |m| = 1, (19h)

where εs, εh, µ, ψ, ψ, θ are design parameters selected later

2Following the remark in Footnote 1, note that the set C=≤(c, c, θ)\{c}is nonempty for all n ≥ 2.

J0

F−1

F1

F0

Obstacle

0 p1

p−1

θ

ε

εh

c

B‖c/2‖(c/2)

B‖µc‖(µc)

εs

J1

J−1

θmax

ε

L(c, p1 − c)

L(c, p−1 − c)

C=≤(c, c, θ)

ψ

ψ

C=≤(c, p1 − c, ψ)

C=≤(c, p−1 − c, ψ)

Fig. 2. 2D illustration of flow and jump sets considered in Sections IV-V.

as in Assumption 2. Before we state our main result, wemotivate the above construction of flow and jump sets.

During the stabilization mode m = 0, the closed-loopsystem should not flow when x is close enough to the surfaceof the obstacle region Bε(c) and the vector field −k0x pointsinside Bε(c). Indeed, by computing the derivative of ‖x−c‖2along solutions to x = −k0x, we can obtain the set wherethe stabilizing vector field −k0x causes a decrease in thedistance ‖x− c‖2 to the centre of the obstacle region Bε(c).This set is characterized by the inequality

−k0x>(x− c) ≤ 0⇐⇒ ‖x− c/2‖2 ≥ ‖c/2‖2 . (20)

The closed set in (20) corresponds to the region outside theball B‖c/2‖(c/2). Therefore, to keep the vehicle safe duringthe stabilization mode, we define around the obstacle thehelmet region H(c, ε, εs, 1/2) used as the jump set J0 in(19c). In other words, if during the stabilization mode thevehicle hits this safety helmet, then the controller jumps toavoidance mode. The amount εs− ε represents the thicknessof the safety helmet defining the jump set J0.

During the avoidance mode |m| = 1, we want our con-troller to slide in the helmet H(c, ε, εh, µ) while maintaininga constant distance to the center c. Note that, with εh > εs

and µ < 1/2, the helmet H(c, ε, εh, µ) (see also Fig. 1)is an inflated version of the helmet H(c, ε, εs, 1/2) andcreates a hysteresis region useful to prevent infinitely manyconsecutive jumps (Zeno behavior). Let us then characterizein the next lemma the equilibria of the avoidance vector fieldκ(x,m) = −kmπ⊥(x− c)(x− pm), |m| = 1.

Lemma 3 ([15]): For each x ∈ Rn\{c} and m ∈ {−1, 1},π⊥(x− c)(x− pm) = 0 if and only if x ∈ L(c, pm − c).Since we want the solutions to leave the set Fm during theavoidance mode, it is necessary to select the point pm andthe flow set Fm such that L(c, pm − c) ∩ Fm = ∅ for eachm ∈ {−1, 1}, otherwise solutions can stay in the avoidancemode indefinitely. This motivates both the intersection withthe conic region in (19e) and Lemma 4, in view of whichwe pose the next assumption.

Assumption 2: The parameters in (19) are selected as:

εh ∈(ε,√ε‖c‖

)εs ∈ (ε, εh) µ ∈ (µmin, 1/2) (21)

θ ∈ (0, θmax) ψ ∈ (0, ψmax) ψ ∈ (ψ,ψmax) (22)

where µmin, θmax and ψmax are defined as

µmin :=1

2

ε2h + ‖c‖2 − 2ε‖c‖‖c‖2 − ε‖c‖

∈ (0, 1/2), (23)

θmax := arccos

(ε2h + ‖c‖2(1− 2µ)

2ε‖c‖(1− µ)

)∈ (0, π/2), (24)

ψmax := min(θ, π/2− θ) ∈ (0, π/4). (25)The intervals in (21)–(25) are well defined. They can bechecked in this order. The intervals of εh and εs are welldefined by Assumption 1. Then, those of µmin, µ, θmax

(θmax > 0 directly from µ > µmin), θ, ψmax and, finally,those of ψ and ψ (equivalent to 0 < ψ < ψ < ψmax) arealso well defined.

Lemma 4 ([15]): Under Assumption 2, Fm ∩ L(c, pm −c) = ∅, for m ∈ {−1, 1}.

V. MAIN RESULT

In this section, we state and prove our main result, whichcorresponds to the objectives discussed in Section III. Wefirst write compactly the flow/jump sets and maps of (19):

F :=⋃

m∈{−1,0,1}

Fm × {m}, J :=⋃

m∈{−1,0,1}

Jm × {m} (26)

(x,m) 7→ F(x,m) := (κ(x,m), 0), (27)(x,m) 7→ J(x,m) := (x,M(x,m)). (28)

The mild regularity conditions satisfied by the hybrid sys-tem (19), as in the next lemma, guarantee the applicabilityof many results in the proof of our main result.

Lemma 5 ([15]): The hybrid system (F ,F,J ,J) satisfiesthe hybrid basic conditions in [16, Ass. 6.5].Let us define the obstacle-free set K and the attractor A as:

K := Rn\Bε(c)× {−1, 0, 1}, A := {0} × {0}. (29)

Our main result is given in the next theorem.Theorem 1: Consider the hybrid system (19) under As-

sumptions 1-2. Then,

Set to which x belongs TF0(x)

∂Bε(c) ∩ B◦‖c/2‖(c/2) P≥(0, x− c)∂Bεs (c)\B‖c/2‖(c/2) P≥(0, x− c)(∂B‖c/2‖(c/2) ∩ B◦εs (c))\Bε(c) P≤(0, x− c/2)∂Bε(c) ∩ ∂B‖c/2‖(c/2) P≥(0, x− c) ∩ P≤(0, x− c/2)∂B‖c/2‖(c/2) ∩ ∂Bεs (c) P≥(0, x− c) ∪ P≤(0, x− c/2)

Set to which x belongs TFm (x)

∂Bε(c)\B‖µc‖(µc)\C≤≤(c, pm−c, ψ) P≥(0, x− c)

∂Bεh(c)\B‖µc‖(µc)\C≤≤(c, pm−c, ψ) P≤(0, x− c)

∂B‖µc‖(µc) ∩ B◦εh (c)\Bε(c) P≥(0, x− µc)C=≤(c, pm − c, ψ) ∩ B

◦εh

(c)\Bε(c) P≥(0, nm(x))

∂Bε(c) ∩ ∂B‖µc‖(µc) P≥(0, x−c)∩P≥(0, x− µc)∂Bεh (c) ∩ ∂B‖µc‖(µc) P≤(0, x−c)∩P≥(0, x− µc)∂Bε(c) ∩ C=≤(c, pm − c, ψ) P≥(0, x−c)∩P≥(0, nm(x))

∂Bεh (c) ∩ C=≤(c, pm − c, ψ) P≤(0, x−c)∩P≥(0, nm(x))

TABLE ITANGENT CONES TO F0 AND Fm AT x, WITH m EITHER −1 OR 1

(nm(x) := πψ(pm − c)(x− c)).

i) all maximal solutions do not have finite escape times,are complete in the ordinary time direction, and theobstacle-free set K in (29) is forward invariant (asin [19, Def. 3.3]);

ii) the set A in (29) is globally asymptotically stable;iii) for each ε′ > ε, it is possible to tune the hybrid con-

troller parameters so that the resulting hybrid feedbacklaw matches, in Rn\Bε′(c), the law u = −k0x.

Theorem 1 shows that the three objectives discussed inSection III are fulfilled.

A. Proof of Theorem 1

To prove item i), we resort to [19, Thm. 4.3]. We firstestablish for H in (19) the relationships invoked in [19,Thm. 4.3], and we refer the reader to Fig. 2 for a two-dimensional visualization. In particular, the boundaries of theflow sets F0 and Fm,m ∈ {−1, 1}, are

∂F0 =(∂Bε(c) ∩ B‖c/2‖(c/2)

)∪(∂Bεs(c)\B‖c/2‖(c/2)

)∪((∂B‖c/2‖(c/2) ∩ Bεs(c))\Bε(c)

), (30)

∂Fm=((∂Bε(c) ∪ ∂Bεh(c))\B‖µc‖(µc)\C≤≤(c, pm−c, ψ)

)∪((∂B‖µc‖(µc)∪C=≤(c, pm−c, ψ))∩Bεh(c)\B◦ε (c)

). (31)

The tangent cone (see [16, Def. 5.12 and Fig. 5.4]), evaluatedat the boundary of these sets, is given in Table I.

Consider m = 0 and let z := κ(x, 0) = −k0x. If x ∈∂Bε(c)∩B◦‖c/2‖(c/2), then (x− c)>z = −k0x>(x− c) > 0

(since x ∈ B◦‖c/2‖(c/2), see (20)), i.e., z ∈ P>(0, x − c).If x ∈ (∂B‖c/2‖(c/2) ∩ B◦εs(c))\Bε(c), then one has (x −c/2)>z = −k0x>(x − c/2) = −k0x>c/2 = −k0‖x‖2/2 ≤0 since x>(x − c) = 0 from ‖x − c/2‖ = ‖c/2‖. Then,z ∈ P≤(0, x − c/2). If x ∈ ∂Bε(c) ∩ ∂B‖c/2‖(c/2) orx ∈ ∂B‖c/2‖(c/2) ∩ ∂Bεs(c), then z>(x − c) = 0 andz>(x − c/2) = −k0‖x‖2/2 ≤ 0 showing, respectively,

that z ∈ P≥(0, x − c) ∩ P≤(0, x − c/2). Finally, if x ∈∂Bεs(c)\B‖c/2‖(c/2), then (x− c)>z = −k0x>(x− c) < 0(since x /∈ B‖c/2‖(c/2)), i.e., z ∈ P<(0, x − c). LetL0 := ∂Bεs(c)\B‖c/2‖(c/2). Therefore, by all the previousarguments, (30) and Table I:

x ∈ L0 =⇒ κ(x, 0) ∩TF0(x) = ∅x ∈ ∂F0\L0 =⇒ κ(x, 0) ⊂ TF0(x).

(32)

Consider then m ∈ {−1, 1} and let now z := κ(x,m) =−kmπ⊥(x − c)(x − pm). If x ∈ ∂Bε(c) or x ∈ ∂Bεh(c)then one has (x − c)>z = −km(x − c)>π⊥(x − c)(x −pm) = 0, which implies that both z ∈ P≥(0, x − c) andz ∈ P≤(0, x − c). Define nm(x) := πψ(pm − c)(x − c),which is a normal vector to the cone C=(c, pm − c, ψ) at x.If x ∈ C=≤(c, pm − c, ψ), then3

nm(x)>z = −kmnm(x)>π⊥(x− c)(x− pm)(3)= km(x− c)>πψ(pm − c)π⊥(x− c)(pm − c)(8),(5)= km(x− c)>(π⊥(pm − c)−sin2(ψ)In)π⊥(x− c)(pm − c)

(3)= km(x− c)>π⊥(pm − c)π⊥(x− c)(pm − c)(5)= km(x− c)>π⊥(pm − c)

(In − π‖(x− c)

)(pm − c)

(3)= −km(x− c)>π⊥(pm − c)π‖(x− c)(pm − c)(1)= −km

(x− c)>π⊥(pm − c)(x− c)‖x− c‖2

(x− c)>(pm − c) ≥ 0

where the last bound follows from π⊥(pm − c) positivesemidefinite and (x−c)>(pm−c) ≤ 0 (since x ∈ C=≤(c, pm−c, ψ) ⊂ P≤(c, pm − c)). Hence, z ∈ P≥(0, nm(x)). Finally,let x ∈ ∂B‖µc‖(µc) ∩ Bεh(c)\B◦ε (c). With θmax in (24) andµ < 1/2, we have

0 ≤c>(c−x)=‖x− c‖2+(1− µ)2‖c‖2−‖x− µc‖2

2(1− µ)

=‖x− c‖2 + ‖c‖2(1− 2µ)

2(1− µ)≤ ε

2h + ‖c‖2(1− 2µ)

2(1− µ)

= cos(θmax)ε‖c‖ ≤ cos(θmax)‖x− c‖‖c‖.

(33)

From c>(pm−c) = − cos(θ)‖c‖‖pm−c‖ (pm ∈ C=≤(c, c, θ)by (18) and Lemma 2) and (33), we get the first bound in

(x− µc)>z = −km(x− µc)>π⊥(x− c)(x− pm)(3)= km(c− µc)>π⊥(x− c)(pm − c)(1)= km(1− µ)(c>(pm − c)

+ c>(c− x) · (x− c)>(pm − c)/‖x− c‖2)

≤ km(1− µ)(− cos(θ) + cos(θmax))‖c‖‖pm − c‖ < 0,

and km > 0, 1 − µ > 0 (from (21)), θ < θmax (from (22))yield the second bound. (x − µc)>z < 0 implies thenz ∈ P<(0, x− µc). Let Lm := ∂B‖µc‖(µc)∩Bεh(c)\B◦ε (c).Therefore, by all the previous arguments, (31) and Table I:

x ∈ Lm =⇒ κ(x,m) ∩TFm(x) = ∅

x ∈ ∂Fm\Lm =⇒ κ(x,m) ⊂ TFm(x).

(34)

3Each (in)equality is obtained thanks to the relationship reported over it.

We can now apply [19, Thm. 4.3]. With K in (29), let F :=∂(K ∩ F)\L with L = ∪m=−1,0,1Lm × {m}. By (32) and(34) and K∩F = F , we have F = ∪m=−1,0,1(∂Fm\Lm)×{m}. It follows from (32) and (34) that for every (x,m) ∈ F ,F(x,m) ⊂ TF (x,m). Also, J(K ∩ J ) ⊂ K, F is closed,the map F satisfies the hybrid basic conditions as provenin Lemma 5 and it is, moreover, locally Lipschitz since itis continuously differentiable. We conclude then that the setK is forward pre-invariant [19, Def. 3.3]. In addition, sinceL0 ⊂ J0 and Lm ⊂ Jm with m ∈ {−1, 1}, one has L ⊂ J .Besides, finite escape times can only occur through flow, andsince the sets F−1 and F1 are bounded by their definitionsin (19e), finite escape times cannot occur for x ∈ F−1 ∪F1.They can neither occur for x ∈ F0 because they wouldmake x>x grow unbounded, and this would contradict thatddt (x

>x) ≤ 0 by the definition of κ(x, 0) and by (19a).Therefore, all maximal solutions do not have finite escapetimes. By [19, Thm. 4.3] again, the set K is actually forwardinvariant [19, Def. 3.3], and solutions are complete. Finally,we anticipate here an immediate corollary of completeness ofsolutions and Lemma 6 below: since the number of jumps isfinite by Lemma 6, all maximal solutions to (19) are actuallycomplete in the ordinary time direction.

To prove item ii), we proceed in two steps. First, we provethat the set A is globally asymptotically stable for the systemwithout jumps. To this end, the jumpless system has dataH 0 = (F,F , ∅, ∅) with flow map F and flow set F definedin (19). We emphasize that H 0 is obtained in accordanceto [20, Eqs. (38)-(39)] by identifying all jumps with events.Consider the Lyapunov function

V(x,m) := m2/2 + ‖x− pm‖2/2, (35)

with p0 := 0 and pm (m ∈ {−1, 1}) defined in (18). One hasV(x,m) = 0 for all (x,m) ∈ A in (29), V(x,m) > 0 forall (x,m) /∈ A, and is radially unbounded relative to F ∪J .Straightforward computations show that

〈∇V(x, 0),F(x, 0)〉 = −k0x>x < 0 ∀x ∈ F0\{0}〈∇V(x,m),F(x,m)〉 = −km(x− pm)>π⊥(x− c)(x− pm)

=−km‖π⊥(x− c)(x− pm)‖2<0 ∀m ∈ {−1, 1}, x ∈ Fm.

The last inequality follows from projection matrices beingpositive semidefinite and Lemma 3, which implies that itcannot be 〈∇V(x,m),F(x,m)〉 = 0 for m ∈ {−1, 1} andall x ∈ Fm since L(c, pm − c) is excluded from Fm byLemma 4. All the above conditions satisfied by V suffice toconclude global asymptotic stability of A for H 0 since Ais compact and H 0 satisfies [16, Ass. 6.5].

Second, the next lemma establishes that the number ofjumps is finite for the given hybrid dynamics in (19).

Lemma 6 ([15]): For H in (19), each solution starting inK experiences no more than 3 jumps.Consequently, global asymptotic stability of A follows fromthe first and second step by [20, Thm. 31], since the hybridsystem in (19) satisfies the Basic Assumptions [20, p. 43],as proven in Lemma 5, the set A is compact and has emptyintersection with the jump set.

Lastly, to prove item iii), let ε′ > ε. Select the parameterεh ∈ (ε,min(ε′,

√ε‖c‖)) while all other hybrid controller

parameters are selected as in Assumption 2. Then this impliesthat the flow sets Fm,m ∈ {−1, 1}, of the avoidance modeare entirely contained in Bε′(c). Therefore, as long as thestate x remains in Rn\Bε′(c), solutions are enforced to flowonly with the stabilizing mode m = 0, which corresponds tothe feedback law u = −k0x.

VI. NUMERICAL EXAMPLE

We illustrate our results through a three-dimensional ex-ample. The hybrid system in (19) is fully specified by thefollowing parameters. The obstacle has center c = (1, 1, 1)and radius ε = 0.700. The controller gains are km = 1 form ∈ {−1, 0, 1}. The parameters used in the constructionof the flow and jump sets are εh = 0.901, εs = 0.800,µ = 0.444, θ = 0.276, which satisfy Assumption 2. Toselect a point p1 ∈ C=≤(c, c, θ)\{c}, we proceed as follows.Select v ∈ Sn such that v>c = 0 and consider R(v, θ) ∈SO(3), i.e., an orthogonal rotation matrix specified by axisv and angle θ. Then, we can verify that the point p1 =(I3−R(v, θ))c is a point on the cone C=≤(c, c, θ). By lettingv = (0, 1,−1), we determine p1 = (0.424,−0.155,−0.155)and p−1 = (−0.348, 0.231, 0.231) as in (18). We also selectψ = 0.249 and ψ = 0.266, which satisfy Assumption 2.Fig. 3 shows that the objectives posed in Section III andproven in Theorem 1 are fulfilled. The top part of thefigure illustrates the relevant sets. The middle part shows thatthe origin is globally asymptotically stable, and the controllaw matches the stabilizing one sufficiently away from theobstacle. The bottom part shows that the solutions are safesince they all stay away from the obstacle set Bε(c).

VII. CONCLUSIONS

We have proposed a hybrid feedback law for the avoidanceof a spherical obstacle in Rn. This law guarantees forwardinvariance of the obstacle-free space and global asymptoticstability of the reference. Future work includes generalizingthis control strategy to multiple, nonspherical obstacles.

REFERENCES

[1] O. Khatib, “Real-time obstacle avoidance for manipulators and mobilerobots,” in Autonomous robot vehicles. Springer, 1986, pp. 396–404.

[2] D. E. Koditschek and E. Rimon, “Robot navigation functions onmanifolds with boundary,” Advances in applied mathematics, vol. 11,no. 4, pp. 412–442, 1990.

[3] D. V. Dimarogonas, S. G. Loizou, K. J. Kyriakopoulos, and M. M.Zavlanos, “A feedback stabilization and collision avoidance schemefor multiple independent non-point agents,” Automatica, vol. 42, no. 2,pp. 229–243, 2006.

[4] I. Filippidis and K. J. Kyriakopoulos, “Navigation functions for focallyadmissible surfaces,” in Amer. Control Conf., 2013, pp. 994–999.

[5] F. Wilson Jr, “The structure of the level surfaces of a Lyapunovfunction,” Journal of Differential Equations, 1967.

[6] S. G. Loizou, “The navigation transformation,” IEEE Trans. Robot.,vol. 33, no. 6, pp. 1516–1523, 2017.

[7] S. Prajna, A. Jadbabaie, and G. J. Pappas, “A framework for worst-case and stochastic safety verification using barrier certificates,” IEEETrans. Automat. Contr., vol. 52, no. 8, pp. 1415–1428, 2007.

[8] P. Wieland and F. Allgower, “Constructive safety using control barrierfunctions,” IFAC Proceedings, vol. 40, no. 12, pp. 462–467, 2007.

[9] M. Z. Romdlony and B. Jayawardhana, “Stabilization with guaranteedsafety using control Lyapunov–barrier function,” Automatica, vol. 66,pp. 39–47, 2016.

2

1.5

1

0.5

2.5

2.5

0

Fig. 3. Top left: sets F−1 (green) and J0 (red) surrounding Bε(c)(grey). Top center: sets J0 (red), J−1 ∩ H(c, ε, εh, µ) (green), andJ1 ∩ H(c, ε, εh, µ) (blue) surrounding Bε(c) (grey). Top right: sets F1

(blue) and J0 (red) surrounding Bε(c) (grey). Middle: phase portrait ofsolutions with different initial conditions and Bε(c) (grey). Bottom: distanceto the obstacle for the solutions and radii εs, ε of H(c, ε, εs, 1/2), Bε(c).

[10] A. D. Ames, X. Xu, J. W. Grizzle, and P. Tabuada, “Control barrierfunction based quadratic programs for safety critical systems,” IEEETrans. Automat. Contr., vol. 62, no. 8, pp. 3861–3876, 2017.

[11] R. G. Sanfelice, M. J. Messina, S. E. Tuna, and A. R. Teel, “Robusthybrid controllers for continuous-time systems with applications toobstacle avoidance and regulation to disconnected set of points,” inAmer. Control Conf., 2006, pp. 3352–3357.

[12] J. I. Poveda, M. Benosman, A. R. Teel, and R. G. Sanfelice, “A hybridadaptive feedback law for robust obstacle avoidance and coordinationin multiple vehicle systems,” in Amer. Control Conf., 2018, pp. 616–621.

[13] P. Braun, C. M. Kellett, and L. Zaccarian, “Unsafe point avoidance inlinear state feedback,” IEEE Conf. on Decision and Control, 2018.

[14] C. D. Meyer, Matrix analysis and applied linear algebra. SIAM, 2000.[15] S. Berkane, A. Bisoffi, and D. V. Dimarogonas, “A hybrid con-

troller for obstacle avoidance in an n-dimensional Euclidean space,”arXiv:1903.04392 [cs.SY], 2019.

[16] R. Goebel, R. G. Sanfelice, and A. R. Teel, Hybrid DynamicalSystems: modeling, stability, and robustness. Princeton UniversityPress, 2012.

[17] L. Wang, A. D. Ames, and M. Egerstedt, “Safety barrier certificatesfor collisions-free multirobot systems,” IEEE Trans. Robot., vol. 33,no. 3, pp. 661–674, 2017.

[18] V. J. Lumelsky and T. Skewis, “Incorporating range sensing in therobot navigation function,” IEEE Trans. Syst., Man, Cybern., vol. 20,no. 5, pp. 1058–1069, 1990.

[19] J. Chai and R. G. Sanfelice, “Forward invariance of sets for hybriddynamical systems (Part I),” IEEE Trans. Automat. Contr., 2019.

[20] R. Goebel, R. G. Sanfelice, and A. R. Teel, “Hybrid dynamicalsystems,” IEEE Control Syst. Mag., vol. 29, no. 2, pp. 28–93, 2009.